48¿ P H I L O S O P H IÆ N A T U R A L I S
D e Mondi erit partium 11163734, & fingulis horis area illa erit partium coógii
S y s t e m a t e t n ■ 1 • • . . 4 * bin latus rectum majus lit vel minus in ratione quavis, erit area di-
urna & horaria major vel minor in eadem ratione fubduplicata.
L E M M A V .
Invenire line am curvam generis parabolici, quce per data
quotcunque puncla tranfibit.
Sunto punfta illa A, B, C, T>, E, F, &c. & ab iifdem ad reftam
quamvis pofitione datam H N demitte perpendicula quotcunque
A H ,, B I , C K , D E , EM , F N.
Caf. i. Si punftorum H, I, K, L, M, N æqualia funt intervalla
H I , I K , K L , &c. collige perpendiculorum A H , B I, C K, &c.
differentias primas b, xb, 3b, 4b, yb, &c. fecundas c, xc, 3c, 4c, &c.
tertias d, xd, 3d, &c. id eli, ita ut fit A H — B I~ b , B I— CK— xb,
C K —D L = ìb ,D L -\ -EM — ^b,— EM-\-FN=z¡ b, &c. dein b—.
x b— c, &c. & fic pergatur ad differentiam ùltimam, qu® hic eft/.
Deinde erefta quacunque perpendiculari RS, qu® fuerit ordinatim
applicata ad curvam qu®fitam: ut inveniatur hujus longitudo, pone
intervalla H I, IK, K L , L M, &c. unitàtes effe, & die À H = a ,
—H S—p, 4 / in—I S = q , ~q in4 -J7<T= r, 4 r in -f- S L = s, 4 s in
- \ - S M = f , pergendo videlicet ad ufque penultimum perpendicu-
lum M E , & pr®ponendo figna negativa terminis HS, IS , &c. qui
jacent
jacent ad partes punfii J verfus A, Se figna affirmativa terminis SK,
SLs &c. qui jacent ad altéras partes punfti S. Et fignis probe ob-
fervatis, erit RS^a-J-bpf -cqf -dr^-esAr f t , &c.
Caf.x. Quod fi punftorum H, I, K, L , &c. inæqualia fint inter-
valla H I , I K , &c. collige perpendiculorum A H , B I , C K , &c.
differentias primas per intervalla perpendiculorum divifas b, xb, 3 b,
4 b, 5 b ; fecundas per intervalla bina diviia c, xc, 3 c, 4 c, &c. tertias
per intervalla terna divifas d, xd, 3 d, See. quartas per intervalla
quaterna divifas e, xe, &c. & fic deinceps ; id eft, ita ut fit b —
A H — B I I B I — C K , „_ C K -T > L ■ f _ F - x b
HI ’ l * = J K ~ ~ * 3 — K L ’ &C' ein H K ’
■xb— 3 b 3 b — 4 b „ n j c — r c j ic— '}C
* c ^ ~ T£ ~ , ^c - K A h &C' P° f e a ^ - H L ' X d ^ -TW - ,
&c. Inventis differentiis, die A H — a, — HS=p, f in — IS ^ q , q:
in + J A = r , r m + S L = s , s in + SM ^ t ; -porgendo feilicet ad ufque
perpendiculum penultimum M E , & erit ordinatim applicata RS-x
a4-bp4- c q 4“ dr4- es 4~f t, See.
Corol. Hinc are® curvarum omnium inveniri poffunt quamproxi-
me. Nam fi curvæ cujufvis quadrandæ inveniantur punfta aliquot,
& parabola per eadem duci intelligatur : erit area parabol® hujus
eadem quamproxime cum area curvæ illius quadrandæ. Potei!;
autem parabola per methodos notiffimas femper quadrati Geome-
trice.
L E M M A VI.
E x obfervatis aliquot locis cometæ invenire locum ejus ad
tempus quodvis intermedium datum.
Defignent H I, IK , K L , L M tempora inter obfervationes ( in
fig. præced.) H A , IB , ICC, L D , M E obfervatas quinque longi-
tudines cometæ, H S tempus datum inter obfervationem primam &
longitudinem quæfitam. Et fi per punfta A, B, C, D, E duci intelligatur
curva regularis A B CT> E-, & per lemma fuperius inveniatur
ejus ordinatim applicata RS, erit RS longitudo quæfita.
Eadem methodo ex obfervatis quinque latitudinibus invenitur. la-
titudo ad tempus. datum.